BinomialAmericanIV

Updated 2023-11-16 15:37:16.437000

Syntax

SELECT [westclintech].[wct].[BinomialAmericanIV] (
  <@CallPut, nvarchar(4000),>
 ,<@AssetPrice, float,>
 ,<@StrikePrice, float,>
 ,<@TimeToMaturity, float,>
 ,<@RiskFreeRate, float,>
 ,<@DividendRate, float,>
 ,<@Price, float,>
 ,<@NSteps, int,>)

Description

Use BinomialAmericanIV to calculate the implied volatility of an American option using the Binomial Tree option pricing formula.

Arguments

@Price

the observed price of the option. @Price is an expression of type float or of a type that can be implicitly converted to float.

@nSteps

the number of steps in the binomial tree. @nSteps is an expression of type int or of a type that can be implicitly converted to int.

@DividendRate

the annualized, continuously compounded dividend rate over the life of the option. For currency options, @Div idendRate should be the foreign risk-free interest rate. @DividendRate is an expression of type float or of a type that can be implicitly converted to float.

@RiskFreeRate

the annualized, continuously compounded risk-free rate of return over the life of the option. @RiskFreeRate is an expression of type float or of a type that can be implicitly converted to float.

@TimeToMaturity

the time to expiration of the option, expressed in years. @Time ToMaturity is an expression of type float or of a type that can be implicitly converted to float.

@CallPut

identifies the option as being a call ('C') or a put ('P'). @CallPut is an expression of type nvarchar or of a type that can be implicitly converted to nvarchar.

@StrikePrice

the exercise price of the option. @Strike Price is an expression of type float or of a type that can be implicitly converted to float.

@AssetPrice

the price of the underlying asset. @AssetPrice is an expression of type float or of a type that can be implicitly converted to float.

Return Type

float

Remarks

@TimeToMaturity must be greater than zero (@TimeToMaturity > 0).

@AssetPrice must be greater than zero (@AssetPrice > 0).

@StrikePrice must be greater than zero (@StrikePrice > 0).

@nSteps must be greater than 1 (@nSteps > 1)

If @DividendRate is NULL an error will be returned

If @RiskFreeRate is NULL an error will be returned

To calculate the price or Greeks of an American option use BJERKSUNDSTENSLAND or BINOMIALAMERICAN.

To calculate the implied volatility of an American option using the Bjerksund & Stensland 2002 formula, use BJERKSUNDSTENSLANDIV.

To calculate the implied volatility on European options use BLACKSCHOLESMERTONIV or BINOMIALEUROIV.

Examples

Calculate the implied volatility for a call option on 2012-09-04, expiring on 2012-12-15, with a current asset price of 99.5, a strike price of 100 and a volatility of 20%. The observed price is 4.1547 and the dividend rate is 0.5%. The number of steps is 100.

SELECT ROUND(wct.BinomialAmericanIV('C', --PutCall
                                    99.5, --Asset Price
                                    100, --Strike Price
                                    datediff(d, '2012-09-04', '2012-12-15') / 365.0000, --Time-to-expiry
                                    .02, --Risk Free Rate
                                    .005, --Dividend Rate
                                    4.1547, --Price
                                    100 --Number of Steps
             ),
             5) as Volatility;

This produces the following result.

{"columns":[{"field":"Volatility","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"Volatility":"0.2"}]}

In the following SELECT, we calculate the implied volatility for a series of strike prices on EUR/USD option which expire in 7 days time. The option price is the midpoint of the bid and ask and the 7 day EURIBOR rate is 0.0335700% and the 7 day LIBOR rate is 0.1870000%.

SELECT CASE
            WHEN call_bid IS NULL THEN NULL
            WHEN call_ask IS NULL THEN NULL
            ELSE wct.BinomialAmericanIV('C',
                                        127.91, --Asset price    
                                        n.s, --Strike   
                                        Cast(7 as float) / cast(365 as float), --Time
                                        0.000335699, --Continuously Compouned EURIBOR
                                        0.001869966, --Continuously Compouned LIBOR
                                        (call_bid + call_ask) / 2, --Price
                                        100 --Number of Steps
                 ) END as [Call IV],
       cast(n.call_bid as money) as Bid,
       cast(n.call_ask as money) as Ask,
       n.s as Strike,
       cast(n.put_bid as money) as Bid,
       cast(n.put_ask as money) as Ask,
       CASE
            WHEN put_bid IS NULL THEN NULL
            WHEN put_ask IS NULL THEN NULL
            ELSE wct.BinomialAmericanIV('P',
                                        127.91, --Asset price
                                        n.s, --Strike
                                        Cast(7 as float) / cast(365 as float), --Time
                                        0.000335699, --Continuously Compouned EURIBOR
                                        0.001869966, --Continuously Compouned LIBIBOR
                                        (put_bid + put_ask) / 2, --Price
                                        100 --Number of Steps
                 ) END as [Put IV]
  FROM (   SELECT 6.85,
                  7.00,
                  121,
                  NULL,
                  .04
           UNION ALL
           SELECT 5.85,
                  6.00,
                  122,
                  NULL,
                  .04
           UNION ALL
           SELECT 4.85,
                  5.00,
                  123,
                  NULL,
                  .04
           UNION ALL
           SELECT 3.85,
                  4.00,
                  124,
                  NULL,
                  .04
           UNION ALL
           SELECT 2.91,
                  2.99,
                  125,
                  .02,
                  .05
           UNION ALL
           SELECT 2.00,
                  2.06,
                  126,
                  .10,
                  .13
           UNION ALL
           SELECT 1.20,
                  1.25,
                  127,
                  .29,
                  .32
           UNION ALL
           SELECT 0.59,
                  0.62,
                  128,
                  .68,
                  .71
           UNION ALL
           SELECT 0.23,
                  0.25,
                  129,
                  1.31,
                  1.35
           UNION ALL
           SELECT 0.06,
                  0.09,
                  130,
                  2.13,
                  2.19
           UNION ALL
           SELECT 0.01,
                  .04,
                  131,
                  3.05,
                  3.15) n(call_bid, call_ask, s, put_bid, put_ask);